Theoretical and experimental studies of Berry and Pancharatnam phases are reviewed. Basic elements of differential geometry are presented for understanding the topological nature of these phases. The basic theory analyzed by Berry in relation to magnetic monopoles is presented. The theory is generalized to nonadiabatic processes and to noncyclic Pancharatnam phases. Different systems are discussed including polarization optics, n-level atomic systems, neutron interferometry and molecular topological phases. |